Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Smooth.of_smooth_fiberToSpecResidueField

∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.LocallyOfFinitePresentation f]
  [AlgebraicGeometry.Flat f],
  (∀ (y : ↥Y), AlgebraicGeometry.Smooth (AlgebraicGeometry.Scheme.Hom.fiberToSpecResidueField f y)) →
    AlgebraicGeometry.Smooth f

If f : X ⟶ Y is locally of finite presentation, flat and has smooth fibers, then f is smooth.

Defined in: Mathlib.AlgebraicGeometry.Morphisms.SmoothFiber
Cited by: 0 results in Mathlib
Foundations: Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.LocallyOfFinitePresentation AlgebraicGeometry.Flat

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